Equivalent Environments and Covering Spaces for Robots

TL;DR

This work develops a rigorous topological framework for when different environments are indistinguishable to a robot with limited sensing and actuation. It introduces history information spaces, sensorimotor structures, and path actions to model robot-environment interactions, and uses covering spaces to show that environment indistinguishability can be witnessed by lifting sensorimotor structure along coverings. A central result links sufficiency via covering maps with necessity via bisimulation, unifying loop closure, SLAM, and graph-exploration phenomena under a single theory. By focusing on geometry- and topology-preserving sensor mappings, the paper explains when local measurements suffice to identify environments and characterizes strong indistinguishability through bisimulation and related concepts. The framework also identifies open questions about complexity, conditions on control-space connectivity, and extensions to non-well-behaved sensors, offering a broad, mathematically grounded view of loop closure and environment inference.

Abstract

This paper formally defines a robot system, including its sensing and actuation components, as a general, topological dynamical system. The focus is on determining general conditions under which various environments in which the robot can be placed are indistinguishable. A key result is that, under very general conditions, covering maps witness such indistinguishability. This formalizes the intuition behind the well studied loop closure problem in robotics. An important special case is where the sensor mapping reports an invariant of the local topological (metric) structure of an environment because such structure is preserved by (metric) covering maps. Whereas coverings provide a sufficient condition for the equivalence of environments, we also give a necessary condition using bisimulation. The overall framework is applied to unify previously identified phenomena in robotics and related fields, in which moving agents with sensors must make inferences about their environments based on limited data. Many open problems are identified.

Figures (8)

• Figure 1: Map reconstruction before loop closure (left) and after loop closure (right). Note how a new metric has been defined on the space to support the new information that the distance of two previously distinct points equals now to zero. Figure reproduced from williams08.
• Figure 2: The spaces on the bottom are strongly indistinguishable from the ones above them. The 2-manifold (a) is a covering space of (d), the 1-complex (b) is a covering space of (e) and (f), whereas (c) is the universal covering space of (b), (e), and (f). For a robot, which senses the local homeomorphism type, (b), (c), (e), and (f) are mutually (not strongly) indistinguishable. Figures are reproduced from AT.
• Figure 3: A robot, which senses the local homeomorphism type, can distinguish between these spaces, which implies that they cannot have a common covering space (see Example \ref{['ex:ApplicationToTop']}). This constitutes an application of the present theory of mathematical robotics to topology.
• Figure 4: The visible star-convex region (left) and the outcome of sensor filtering (right). The filter is a topological invariant of the star-region as we will see in Section \ref{['sec:Revisiting']}. Image reproduced from TovMurLav07
• Figure 5: An example of a sensor mapping which is a metric invariant of the local neighbourhood of the robot. See Section \ref{['sec:Revisiting']} for more details. Image reproduced from Katsev2011

Theorems & Definitions (76)

• Definition 3.1
• Proposition 3.2
• proof
• Remark 3.3
• Definition 3.4: Metric on $\mathcal{U}_M$
• Proposition 3.5
• proof
• Proposition 3.6
• proof
• Proposition 3.7